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A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. [1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.
In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis. For a function f : X → Y {\displaystyle f\colon X\to Y} , the set Y is called the codomain : the set to which all outputs must belong.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain of function, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. There may be a number of domain elements which map to the same range element.
In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter. [nb 1] Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g.
Domain coloring plot of the function f(x) = (x 2 − 1)(x − 2 − i) 2 / x 2 + 2 + 2i , using the structured color function described below.. In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane.
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One can define the cokernel in the general framework of category theory.In order for the definition to make sense the category in question must have zero morphisms.The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0 XY : X → Y.